Wrapping geometry object animation
I suggest using an armature to animate the wrapping.

Animating a rotation with shape keys will lead to deformed faces during the animation as the shape key only considers the location of the vertices. Between two keyframes the location is interpolated on a straight line.
Displayed below are two planes which get rotated by 45 degrees. The red plane is rotated by entering the angle in the transformation field while the blue plane uses a shape key with rotated vertices.



By using an armature you can avoid this deformation.
The bones' heads should be placed at the dodecahedron's edges and the tails at the center of the faces. I assume the best way to achieve this is to alternate using Shift+S > Cursor to Selection and Selection to Cursor.



The default rotation mode for bones is Quaternion. You can switch to one of the Euler angle modes to get the correct rotation axis. Gimbal lock should be no problem in this case as you're rotating on only one axis.